2009
DOI: 10.1016/j.jfa.2009.01.028
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Banach spaces without minimal subspaces

Abstract: We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of s… Show more

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Cited by 40 publications
(160 citation statements)
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“…This was further refined in [3] and, in [14], was formulated as the determinacy of two related adversarial Gowers games, A X and B X .…”
Section: Introductionmentioning
confidence: 99%
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“…This was further refined in [3] and, in [14], was formulated as the determinacy of two related adversarial Gowers games, A X and B X .…”
Section: Introductionmentioning
confidence: 99%
“…For applications of the above dichotomies to the geometry of Banach spaces, we refer the reader to [6], [3] and [13]. Acknowledgement: The author is grateful for a number of insightful comments and useful discussions with A. Montalban, J. Moore and P. Welch on the topic of this paper.…”
Section: Introductionmentioning
confidence: 99%
“…Relaxing this notion one obtains quasiminimality, which asserts that any two infinitely dimensional subspaces of a given space contain further two isomorphic infinitely dimensional subspaces. Relaxing the notions of minimality or adding additional requirements of choice of isomorphic subspaces in quasi-minimality case leads to different types of minimality of a space, contrasted in [16,10] with different types of tightness, categorized in [10].…”
Section: Introductionmentioning
confidence: 99%
“…(6) No disjointly supported subspaces of X wh are isomorphic. Using the terminology from [14] the space X wh is tight by support. In the above and herein, we use L(Y ) to denote the Banach space of bounded linear operators on Y .…”
Section: Introductionmentioning
confidence: 99%
“…In the next section we give a description of the norm of X wh and further discuss some of its critical properties. We note that in [14] they construct a space that is strongly asymptotic 1 and has a property they call tight by support. The space X wh is the first example of a strongly asymptotic 2 space that is tight by support.…”
Section: Introductionmentioning
confidence: 99%